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\subsection{The differential}
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\compactdef{Differentiable function} We have function $f: X \rightarrow \R^m$, linear map $u : \R^n \rightarrow \R^m$ and $x_0 \in X$. $f$ is differentiable at $x_0$ with differential $u$ if
$\displaystyle \lim_{\elementstack{x \rightarrow x_0}{x \neq x_0}} \frac{1}{||x - x_0||} (f(x) - f(x_0) - u(x - x_0) = 0$ where the limit is in $\R^m$.
We denote $\dx f(x_0) = u$.
If $f$ is differentiable at every $x_0 \in X$, then $f$ is differentiable on $X$

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\shortproposition Let $f, g : X \rightarrow \R^m$ with $X \subseteq \R^n$ open
\begin{itemize}[noitemsep]
    \item The function $f + g$ is differentiable with differential $\dx (f + g) = \dx f + \dx g$
\end{itemize}